Parametric Equations

If  $x=e^{-t}$  and  $y=\cos\left(2t\right)$  then $\frac{dy}{dx}=$  

A particle moves on a plane curve so that at any time t > 0 its x-coordinate is  $t^3-t$  and its y-coordinate is  $\left(2t-1\right)^3$  . The acceleration vector of the particle at t = 1 is

The length of the path described by the parametric equations  $x=\frac{1}{3}t^3$  and  $y=\frac{1}{2}t^2$  where  $0\le t\le1$  is given by 

A curve C is defined by the parametric equations  $x=t^2-4t+1$  and  $y=t^3$  . Which of the following is an equation of the line tangent to the graph of C at the point (-3,8)?

Find  $\frac{d^2y}{dx^2}$  for the curve given by  $x=t^2+1$  and  $y=t^3$  .

If  $x=\cos t$  and  $y=2\sin^2t$  then  $\frac{dy}{dx}$  at t = 1 is

For what value(s) of t does the curve defined by the parametric equations  $x=\frac{5t}{1+t^3}$  and  $y=\frac{2t^2}{1+t^3}$  have a horizontal tangent?

An object moving along a curve in the xy-plane is in position  $\left(x\left(t\right),y\left(t\right)\right)$  at time  $t\ge0$  with  $\frac{dx}{dt}=2-\sin\left(t^2\right)$  . At time t = 3, the object is at position (2,7). What is the x coordinate of the position of the object at time t = 6? 

A curve in the plane is defined parametrically by the equations  $x=t^3+t$  and  $y=t^4+2t^2$  . An equation of the line tangent to the curve at t = 1 is 

Find the point on the graph of $x=t^2-2$  and  $y=-t+2$  when  $t=-2$ .